We study the edge transport properties of 2d interacting Hall systems, displaying singlemode chiral edge currents. For this class of many-body lattice models, including for instance the interacting Haldane model, we prove the quantization of the edge charge conductance and the bulk-edge correspondence. Instead, the edge Drude weight and the edge susceptibility are interaction-dependent; nevertheless, they satisfy exact universal scaling relations, in agreement with the chiral Luttinger liquid theory. Moreover, charge and spin excitations differ in their velocities, giving rise to the spin-charge separation phenomenon. The analysis is based on exact renormalization group methods, and on a combination of lattice and emergent Ward identities. The invariance of the emergent chiral anomaly under the renormalization group flow plays a crucial role in the proof. phases [35]. We prove the exact quantization of the edge conductance, for weak interactions: all interaction corrections cancel out. Combined with [27] and with the noninteracting bulk-edge correspondence [39,51,16], this result provides the first proof of the bulk-edge correspondence for an interacting many-body quantum system. Moreover, we also consider the edge Drude weight and the edge susceptibility, both for charge and spin degrees of freedom; we find explicit expressions for these quantities, which turn out to be nonuniversal in the coupling strength. Nevertheless, the Drude weight D and the susceptibility κ satisfy the universal scaling relation D " κv 2 c , as in the Luttinger model. Finally, we compute the two-point function, and we show that it exhibits spin-charge separation.Notice that our analysis does not extend in a straightforward way to the case of multi-edge currents. The reason being the scattering between different edge modes. In the renormalization group terminology, the edge states scattering is a marginal process; our method allows to control the scattering between edge states with the same velocity, thanks to the comparison with the chiral Luttinger model (see below), but does not allow to control the scattering of edge states with different velocities. We leave the generalization to multi-edge channels Hall systems as a very interesting open problem, on which we plan to come back in the future.The paper is organized as follows. In Section 2 we introduce the class of interacting lattice models we will consider, and we define bulk and edge transport coefficients, in the linear response regime. In Section 3 we recall some known facts about noninteracting Hall systems. Then, in Section 4 we present our main result, Theorem 4.1. In the rest of the paper, we discuss the proof of Theorem 4.1. In Section 5 we introduce a functional integral representation for fermionic lattice models, and in particular in Section 5.2 we derive a rigorous relationship between the model of interest and an interacting one-dimensional quantum field theory. This result actually applies to models with a general number of edge states. Starting from Section 6, we restr...

We consider a class of non-integrable, or non-planar, Ising models in two dimensions, whose Hamiltonian, in addition to the standard nearest neighbor couplings, includes additional weak multi-spin interactions which are even under spin flip. We study the model in cylindrical domains of arbitrary aspect ratio and compute the multipoint energy correlations at the critical temperature in terms of a multiscale expansion, uniformly convergent in the domain size and in the lattice spacing. We prove that, in the scaling limit, the multipoint energy correlations converge to the same limiting correlations as those of the planar Ising model in a finite cylinder with renormalized horizontal and vertical couplings, up to an overall multiplicative constant independent of the shape of the domain. The proof is based on a representation of the generating function of correlations in terms of a non-Gaussian Grassmann integral, and a constructive Renormalization Group (RG) analysis thereof.A key technical novelty compared with previous works is a systematic analysis of the effect of the boundary corrections to the RG flow, in particular a proof that the scaling dimension of boundary operators is better by one dimension than their bulk counterparts. In addition, a cancellation mechanism based on an approximate image rule for the fermionic Green's function is of crucial importance for controlling the flow of the (superficially) marginal boundary terms under RG iterations.

In this paper, meant as a companion to Antinucci et al. (Energy correlations of non-integrable Ising models: the scaling limit in the cylinder, 2020. arXiv: 1701.05356), we consider a class of non-integrable 2D Ising models in cylindrical domains, and we discuss two key aspects of the multiscale construction of their scaling limit. In particular, we provide a detailed derivation of the Grassmann representation of the model, including a self-contained presentation of the exact solution of the nearest neighbor model in the cylinder. Moreover, we prove precise asymptotic estimates of the fermionic Green’s function in the cylinder, required for the multiscale analysis of the model. We also review the multiscale construction of the effective potentials in the infinite volume limit, in a form suitable for the generalization to finite cylinders. Compared to previous works, we introduce a few important simplifications in the localization procedure and in the iterative bounds on the kernels of the effective potentials, which are crucial for the adaptation of the construction to domains with boundaries.

In this paper, we study a hierarchical supersymmetric model for a class of gapless, three-dimensional, weakly disordered quantum systems, displaying pointlike Fermi surface and conical intersections of the energy bands in the absence of disorder. We use rigorous renormalization group methods and supersymmetry to compute the correlation functions of the system. We prove algebraic decay of the two-point correlation function, compatible with delocalization. A main technical ingredient is the multiscale analysis of massless bosonic Gaussian integrations with purely imaginary covariances, performed via iterative stationary phase expansions.

We consider a class of non-integrable 2D Ising models whose Hamiltonian, in addition to the standard nearest neighbor couplings, includes additional weak multi-spin interactions which are even under spin flip. We study the model in cylindrical domains of arbitrary aspect ratio and compute the multipoint energy correlations at the critical temperature via a multiscale expansion, uniformly convergent in the domain size and in the lattice spacing. We prove that, in the scaling limit, the multipoint energy correlations converge to the same limiting correlations as those of the nearest neighbor Ising model in a finite cylinder with renormalized horizontal and vertical couplings, up to an overall multiplicative constant independent of the shape of the domain. The proof is based on a representation of the generating function of correlations in terms of a non-Gaussian Grassmann integral, and a constructive Renormalization Group (RG) analysis thereof. A key technical novelty compared with previous works is a systematic analysis of the effect of the boundary corrections to the RG flow, in particular a proof that the scaling dimension of boundary operators is better by one dimension than their bulk counterparts. In addition, a cancellation mechanism based on an approximate image rule for the fermionic Green’s function is of crucial importance for controlling the flow of the (superficially) marginal boundary terms under RG iterations.